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## Cluster structures on quantum coordinate rings, (2013)

Venue: | Selecta Math. (N.S.) |

Citations: | 10 - 0 self |

### Citations

715 |
Introduction to quantum groups
- Lusztig
- 1993
(Show Context)
Citation Context ... Gup ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ )) we get by Lemma 6.1(a) that ϕ(duk(λ ),v(λ )) = (ik e ′)(bk) ( Gup ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ )) and bk = ε∗ik ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞) ) . Thus, applying Lemma 6.2, we get that ϕ(duk(λ ),v(λ )) = G up ( (e∗ik) bk · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ ) , and the statement follows by induction on k. 2 7 Quantum unipotent subgroups In this section we provide a quantum version of the coordinate ring C[N(w)] studied in [GLS6], following [Lu2, Sa, Ki]. 7.1 The quantum enveloping algebra Uq(n(w)) We fix w∈W , and we denote by ∆+w the subset of positive roots α of g such that w(α) is a negative root. This gives rise to a finite-dimensional Lie subalgebra n(w) := ⊕ α∈∆+w nα of n, of dimension l(w). The graded dual U(n(w))∗gr can be identified with the coordinate ring C[N(w)] of a unipotent subgroup N(w) of the Kac-Moody group G with Lie(N(w)) = n(w). (For more details, see [GLS6].) 20 In order to define a q-analogue of U(n(w)), one introduces Lusztig’s braid group operation on Uq(g) [Lu2]. For i ∈ I, Lusztig has proved the existence of a Q(q)... |

405 | On crystal bases of the q-analogue of universal enveloping algebras
- Kashiwara
- 1991
(Show Context)
Citation Context ...s B. If we regard C as an A-module via the homomorphism q 7→ 1, we can define U1(n) := C⊗AUA(n). (6.8) This is a C-algebra isomorphic to the enveloping algebra U(n). Similarly, let AA(n) be the A-submodule of Aq(n) spanned by the basis Ψ(B∗). Define A1(n) := C⊗A AA(n). (6.9) This is a C-algebra isomorphic to the graded dual U(n)∗gr. This commutative ring can be identified with the coordinate ring C[N] of a pro-unipotent pro-group N with Lie algebra the completion n of n (see [GLS6]). 17 6.4 Global bases of Uq(n−) We shall also use Kashiwara’s lower global basis Blow of Uq(n−), constructed in [K1]. It was proved by Grojnowski and Lusztig that ϕ(B) = Blow, where ϕ is the anti-automorphism of (2.1). For i ∈ I, we introduce the q-derivations e′i and ie′ of Uq(n−), defined by e′i( f j) = ie′( f j) = δi j and, for homogeneous elements x,y ∈Uq(n−), e′i(xy) = e ′ i(x)y+q 〈hi,wt(x)〉xe′i(y), (6.10) ie′(xy) = q〈hi,wt(y)〉ie′(x)y+ x ie′(y). (6.11) Note that ie′ = ∗◦e′i ◦∗. Let us denote by (·, ·)K the Kashiwara scalar product on Uq(n−). It is the unique symmetric bilinear form such that (1, 1)K = 1, and ( fix, y)K = (x, e′i(y))K , (x ∈Uq(n−), y ∈Uq(n−), i ∈ I). (6.12) It also satisfies (x fi, y)K ... |

199 |
Perverse sheaves, and quantized enveloping algebras
- Lusztig, Quivers
- 1991
(Show Context)
Citation Context ...quantum enveloping algebra Uq(g). Let n denote the nilpotent subalgebra arising from a triangular decomposition of g. In the case when g is finite-dimensional, Ringel [Ri] showed that the positive part Uq(n) of Uq(g) can be realized as the (twisted) Hall algebra of the category of representations over Fq2 of a quiver Q, obtained by orienting the Dynkin diagram of g. This was a major inspiration for Lusztig’s geometric realization of Uq(n) in terms of Grothendieck groups of categories of perverse sheaves over varieties of representations of Q, which is also valid when g is infinite-dimensional [Lu1]. The constructions of Ringel and Lusztig involve the choice of an orientation of the Dynkin diagram. In an attempt to get rid of this choice, Lusztig replaced the varieties of representations of Q by the varieties of nilpotent representations of its preprojective algebra Λ = Λ(Q), which depends only on the underlying unoriented graph. He showed that one can realize the enveloping algebra U(n) as an algebra of C-valued constructible functions over these nilpotent varieties [Lu1, Lu4]. The multiplication of U(n) is obtained as a convolution-type product similar to the product of the Ringel-Hall... |

144 | Cluster algebras III. Upper bounds and double Bruhat cells
- Berenstein, Fomin, et al.
(Show Context)
Citation Context ...ves PTi . One defines a quiver ΓT with vertex set {1, . . . ,r}, and di j arrows from i to j, where di j = dimExt1AT (STi ,STj). (This is known as the Gabriel quiver of AT .) Most of the information contained in ΓT can be encoded in an r× (r−n)-matrix BT = (bi j)1≤i≤r, 1≤ j≤r−n, given by bi j = (number of arrows j→ i in ΓT )− (number of arrows i→ j in ΓT ). (9.6) Note that BT can be regarded as an exchange matrix, with skew-symmetric principal part. The next theorem gives an explicit description of the quiver ΓT (hence also of the matrix BT ) for certain Cw-maximal rigid modules. Following [BFZ], we define a quiver Γi as follows. The vertex set of Γi is equal to {1, . . . ,r}. For 1≤ k ≤ r, let k− := max({0}∪{1≤ s≤ k−1 |is = ik}) , (9.7) k+ := min({k+1≤ s≤ r |is = ik}∪{r+1}) . (9.8) For 1≤ s, t ≤ r such that is 6= it , there are |ais,it |arrows from s to t provided t+ ≥ s+ > t > s. These are called the ordinary arrows of Γi. Furthermore, for each 1 ≤ s ≤ r there is an arrow s→ s− provided s− > 0. These are the horizontal arrows of Γi. The following result generalizes [GLS3, Theorem 1] (see [GLS4, Theorem 2.3] in the case when w is adaptable). Theorem 9.3 ([BIRS, Theorem II.4.1]) The ... |

142 | PBW-bases of quantum groups
- Ringel
- 1996
(Show Context)
Citation Context ...ystem. In case G is a simple algebraic group of type A,D,E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure. 1 Introduction Let g be the Kac-Moody algebra associated with a symmetric Cartan matrix. Motivated by the theory of integrable systems in statistical mechanics and quantum field theory, Drinfeld and Jimbo have introduced its quantum enveloping algebra Uq(g). Let n denote the nilpotent subalgebra arising from a triangular decomposition of g. In the case when g is finite-dimensional, Ringel [Ri] showed that the positive part Uq(n) of Uq(g) can be realized as the (twisted) Hall algebra of the category of representations over Fq2 of a quiver Q, obtained by orienting the Dynkin diagram of g. This was a major inspiration for Lusztig’s geometric realization of Uq(n) in terms of Grothendieck groups of categories of perverse sheaves over varieties of representations of Q, which is also valid when g is infinite-dimensional [Lu1]. The constructions of Ringel and Lusztig involve the choice of an orientation of the Dynkin diagram. In an attempt to get rid of this choice, Lusztig replaced the va... |

114 | Double Bruhat cells and total positivity
- Fomin, Zelevinsky
- 1999
(Show Context)
Citation Context ...G/P. 2 Note also that, by taking w equal to the square of a Coxeter element, our result gives a Lie theoretic realization of all quantum cluster algebras associated with an arbitrary acyclic quiver (but with a particular choice of coefficients). Our strategy for proving Theorem 1.1 can be summarized as follows. Let Aq(g) be the quantum analogue of the coordinate ring constructed by Kashiwara [K2]. We first obtain a general quantum determinantal identity in Aq(g) (Proposition 3.2). This is a q-analogue (and an extension to the Kac-Moody case) of a determinantal identity of Fomin and Zelevinsky [FZ1]. We then transfer this identity to Aq(n) (Proposition 5.4). Here a little care must be taken since the restriction map from Aq(g) to Aq(n) is not a ring homomorphism. Appropriate specializations of this identity give rise, for every w ∈W , to a system Σw of equations (Proposition 5.5) allowing to calculate certain quantum minors depending on w in a recursive way. This system is a q-analogue of a T -system arising in various problems of mathematical physics and combinatorics (see [KNS]), and we believe it could be of independent interest. It will turn out that all quantum minors involved in Σw... |

108 | Cluster structures for 2-Calabi-Yau categories and unipotent groups
- Buan, Iyama, et al.
(Show Context)
Citation Context ... function ϕX ∈ C[N], and the product ϕX ϕY can be calculated in terms of varieties of short exact sequences with end-terms X and Y [GLS2]. For each element w of the Weyl group W of g, the group N has a finite-dimensional subgroup N(w) of dimension equal to the length of w. In particular, when g is finite-dimensional, we have N = N(w0), where w0 is the longest element of W . In [GLS6], we have shown that the coordinate ring C[N(w)] is spanned by the functions ϕX where X goes over the objects of a certain subcategory Cw of mod(Λ). This category was introduced by Buan, Iyama, Reiten and Scott in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the sense of Fomin and Zelevinsky [FZ2], for which the cluster monomials are of the form ϕT for rigid objects T of Cw. The algebra C[N] has a quantum deformation, which we denote by Aq(n) (see below § 4.2), and it is well known that the algebras Uq(n) and Aq(n) are in fact isomorphic. By works of Lusztig and De Concini-Kac-Procesi, Aq(n) has a subalgebra Aq(n(w)), which can be regarded as a quantum deformation of C[N(w)]. On the other hand, Berenstein and Zelevinsky [BZ2] hav... |

104 |
Lectures on algebraic quantum groups
- Brown, Goodearl
- 2002
(Show Context)
Citation Context ...lusive, although Lampe [La1, La2] has proved it for all cluster variables in the two special cases mentioned above. Finally we note that it is well known that the algebras Aq(n(w)) are skew polynomial rings. Therefore, by Theorem 1.1, all quantum cluster algebras of the form AQ(q)(Cw) are also skew polynomial rings, which is far from obvious from their definition. One may hope that, conversely, the existence of a cluster structure on many familiar quantum coordinate rings will bring some new insights for studying their ring-theoretic properties, a very active subject in recent years (see e.g. [BG, GLL, MC, Y] and references therein). 3 2 The quantum coordinate ring Aq(g) 2.1 The quantum enveloping algebra Uq(g) Let g be a symmetric Kac-Moody algebra with Cartan subalgebra t. We follow the notation of [K2, §1]. In particular, we denote by I the indexing set of the simple roots αi (i ∈ I) of g, by P ⊂ t∗ its weight lattice, by hi(i ∈ I) the elements of P∗ ⊂ t such that 〈hi,α j〉= ai j are the entries of the generalized Cartan matrix of g. Since g is assumed to be symmetric, we also have a symmetric bilinear form (·, ·) on t∗ such that (αi,α j) = ai j. The Weyl group W < GL(t∗) is the Coxeter group ge... |

104 | Quantum groups and their primitive ideals, - Joseph - 1995 |

88 | Rigid modules over preprojective algebras - Geiß, Leclerc, et al. |

82 |
Global crystal bases of quantum groups,
- KASHIWARA
- 1993
(Show Context)
Citation Context ...a cluster structure on the quantum coordinate ring of the unipotent radical NP of P, which can be regarded as a cluster structure on the quantum coordinate ring of an open cell of the partial flag variety G/P. 2 Note also that, by taking w equal to the square of a Coxeter element, our result gives a Lie theoretic realization of all quantum cluster algebras associated with an arbitrary acyclic quiver (but with a particular choice of coefficients). Our strategy for proving Theorem 1.1 can be summarized as follows. Let Aq(g) be the quantum analogue of the coordinate ring constructed by Kashiwara [K2]. We first obtain a general quantum determinantal identity in Aq(g) (Proposition 3.2). This is a q-analogue (and an extension to the Kac-Moody case) of a determinantal identity of Fomin and Zelevinsky [FZ1]. We then transfer this identity to Aq(n) (Proposition 5.4). Here a little care must be taken since the restriction map from Aq(g) to Aq(n) is not a ring homomorphism. Appropriate specializations of this identity give rise, for every w ∈W , to a system Σw of equations (Proposition 5.5) allowing to calculate certain quantum minors depending on w in a recursive way. This system is a q-analogue... |

71 | Quantum cluster algebras
- Berenstein, Zelevinsky
- 2005
(Show Context)
Citation Context ...in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the sense of Fomin and Zelevinsky [FZ2], for which the cluster monomials are of the form ϕT for rigid objects T of Cw. The algebra C[N] has a quantum deformation, which we denote by Aq(n) (see below § 4.2), and it is well known that the algebras Uq(n) and Aq(n) are in fact isomorphic. By works of Lusztig and De Concini-Kac-Procesi, Aq(n) has a subalgebra Aq(n(w)), which can be regarded as a quantum deformation of C[N(w)]. On the other hand, Berenstein and Zelevinsky [BZ2] have introduced the concept of a quantum cluster algebra. They have conjectured that the quantum coordinate rings of double Bruhat cells in semisimple algebraic groups should have a quantum cluster algebra structure. In this paper, we introduce for every w an explicit quantum cluster algebra AQ(q)(Cw), defined in a natural way in terms of the category Cw. In particular, for every reachable rigid object T of Cw there is a corresponding quantum cluster monomial YT ∈ AQ(q)(Cw). Our main result is the following quantization of the above theorem of [GLS6]. Theorem 1.1 There is an algebra isomorphi... |

67 | Semicanonical bases and preprojective algebras II: a multiplication formula,
- GEISS, LECLERC, et al.
- 2007
(Show Context)
Citation Context ...me this difficulty by switching to the dual picture. Keywords: quantum cluster algebra, quantum group, preprojective algebra, quantum minor, T -system Mathematics Subject Classification (2010): 13F60, 16G20, 17B37 1 The dual of U(n) as a Hopf algebra can be identified with the algebra C[N] of regular functions on the pro-unipotent group N attached to n. Dualizing Lusztig’s construction, one can obtain for each nilpotent representation X of Λ a distinguished regular function ϕX ∈ C[N], and the product ϕX ϕY can be calculated in terms of varieties of short exact sequences with end-terms X and Y [GLS2]. For each element w of the Weyl group W of g, the group N has a finite-dimensional subgroup N(w) of dimension equal to the length of w. In particular, when g is finite-dimensional, we have N = N(w0), where w0 is the longest element of W . In [GLS6], we have shown that the coordinate ring C[N(w)] is spanned by the functions ϕX where X goes over the objects of a certain subcategory Cw of mod(Λ). This category was introduced by Buan, Iyama, Reiten and Scott in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the sense o... |

52 |
Semicanonical bases arising from enveloping algebras
- Lusztig
(Show Context)
Citation Context ...ories of perverse sheaves over varieties of representations of Q, which is also valid when g is infinite-dimensional [Lu1]. The constructions of Ringel and Lusztig involve the choice of an orientation of the Dynkin diagram. In an attempt to get rid of this choice, Lusztig replaced the varieties of representations of Q by the varieties of nilpotent representations of its preprojective algebra Λ = Λ(Q), which depends only on the underlying unoriented graph. He showed that one can realize the enveloping algebra U(n) as an algebra of C-valued constructible functions over these nilpotent varieties [Lu1, Lu4]. The multiplication of U(n) is obtained as a convolution-type product similar to the product of the Ringel-Hall algebra, but using Euler characteristics of complex varieties instead of number of points of varieties over finite fields. Note that this realization of U(n) is only available when the Cartan matrix is symmetric. One of the motivations of this paper was to find a similar construction of the quantized enveloping algebra Uq(n), as a kind of Ringel-Hall algebra attached to a category of representations of Λ. Unfortunately, there seems to be no simple way of q-deforming Lusztig’s realiz... |

48 | Cluster algebra structures and semicanonical bases for unipotent groups, arXiv:math/0703039v4 [math.RT
- Geiss, Leclerc, et al.
(Show Context)
Citation Context ...product ϕX ϕY can be calculated in terms of varieties of short exact sequences with end-terms X and Y [GLS2]. For each element w of the Weyl group W of g, the group N has a finite-dimensional subgroup N(w) of dimension equal to the length of w. In particular, when g is finite-dimensional, we have N = N(w0), where w0 is the longest element of W . In [GLS6], we have shown that the coordinate ring C[N(w)] is spanned by the functions ϕX where X goes over the objects of a certain subcategory Cw of mod(Λ). This category was introduced by Buan, Iyama, Reiten and Scott in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the sense of Fomin and Zelevinsky [FZ2], for which the cluster monomials are of the form ϕT for rigid objects T of Cw. The algebra C[N] has a quantum deformation, which we denote by Aq(n) (see below § 4.2), and it is well known that the algebras Uq(n) and Aq(n) are in fact isomorphic. By works of Lusztig and De Concini-Kac-Procesi, Aq(n) has a subalgebra Aq(n(w)), which can be regarded as a quantum deformation of C[N(w)]. On the other hand, Berenstein and Zelevinsky [BZ2] have introduced the concept of a... |

44 |
String bases for quantum groups of type Ar
- Berenstein, Zelevinsky
- 1993
(Show Context)
Citation Context ...β (a))b(a). Proof — This is a restatement in our setup of [Ki, Theorem 4.26]. The same result was previously obtained in [La1] in a particular case. 2 It follows from (10.27) that all quantum cluster monomials satisfy condition (b) of Proposition 12.8. This is similar to [BZ2, Proposition 10.9 (2)]. Unfortunately, it is not easy to prove that quantum cluster monomials satisfy (a), so one can only conjecture Conjecture 12.9 All quantum cluster monomials YR, where R runs over the set of reachable rigid modules in Cw, belong to B(w). It follows from the original work of Berenstein and Zelevinsky [BZ1] that the conjecture holds in the prototypical Example 12.2, namely, for g= sl4 the dual canonical basis of Uq(n) is equal to the set of quantum cluster monomials. In this case, there are 14 clusters and 12 cluster variables (including the frozen ones), which all are unipotent quantum minors. 46 We note that the conjecture is satisfied when R = M[b,a] is one of the modules occuring in the quantum determinantal identities. Indeed, YM[b,a] is then a quantum minor, and belongs to B(w) by Proposition 6.3. It is proved in [La1, La2] that the conjecture holds for all quantum cluster variables when A... |

35 |
PBW basis of quantized universal enveloping algebras,
- SAITO
- 1994
(Show Context)
Citation Context ... Gup ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ )) we get by Lemma 6.1(a) that ϕ(duk(λ ),v(λ )) = (ik e ′)(bk) ( Gup ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ )) and bk = ε∗ik ( (e∗ik−1) bk−1 · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞) ) . Thus, applying Lemma 6.2, we get that ϕ(duk(λ ),v(λ )) = G up ( (e∗ik) bk · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ ) , and the statement follows by induction on k. 2 7 Quantum unipotent subgroups In this section we provide a quantum version of the coordinate ring C[N(w)] studied in [GLS6], following [Lu2, Sa, Ki]. 7.1 The quantum enveloping algebra Uq(n(w)) We fix w∈W , and we denote by ∆+w the subset of positive roots α of g such that w(α) is a negative root. This gives rise to a finite-dimensional Lie subalgebra n(w) := ⊕ α∈∆+w nα of n, of dimension l(w). The graded dual U(n(w))∗gr can be identified with the coordinate ring C[N(w)] of a unipotent subgroup N(w) of the Kac-Moody group G with Lie(N(w)) = n(w). (For more details, see [GLS6].) 20 In order to define a q-analogue of U(n(w)), one introduces Lusztig’s braid group operation on Uq(g) [Lu2]. For i ∈ I, Lusztig has proved the existence of a Q(q)... |

27 |
Regular functions on certain infinite-dimensional groups. Arithmetic and geometry
- Kac, Peterson
- 1983
(Show Context)
Citation Context ...bining (2.6) and (2.10) we obtain x · (ψ θ) · y = ∑(x(1) ·ψ · y(1))(x(2) ·θ · y(2)). (2.11) 2.4 Peter-Weyl theorem Following Kashiwara [K2, §7], we define Aq(g) as the subspace of Uq(g)∗ consisting of the linear forms ψ such that the left submodule Uq(g)ψ belongs to Oint(g), and the right submodule ψUq(g) belongs to Oint(gop). It follows from the fact that the categories Oint(g) and Oint(gop) are closed under tensor product that Aq(g) is a subring of Uq(g)∗. The next proposition of Kashiwara can be regarded as a q-analogue of the Peter-Weyl theorem for the Kac-Moody group G attached to g (see [KP]). We include a proof for the convenience of the reader. Proposition 2.1 ([K2, Proposition 7.2.2]) We have an isomorphism Φ of Uq(g)-bimodules⊕ λ∈P+ V r(λ )⊗V (λ ) ∼−→ Aq(g) given by Φ(n⊗m)(x) = 〈nx, m〉λ , (m ∈V (λ ), n ∈V r(λ ), x ∈Uq(g)). Proof — It follows from (2.5) and (2.6) that Φ defines a homomorphism of Uq(g)-bimodules from ⊕λ∈P+V r(λ )⊗V (λ ) to Uq(g)∗. Since V (λ ) and V r(λ ) are integrable for all λ ∈ P+, we see that ImΦ⊆ Aq(g). Let us show that Φ is surjective. Let ψ ∈ Aq(g). We want to show that ψ ∈ ImΦ. Since V :=Uq(g)ψ is integrable, it decomposes as a (finite) direct sum of i... |

26 |
Partial flag varieties and preprojective algebras,
- GEISS, LECLERC, et al.
- 2008
(Show Context)
Citation Context ...16 non frozen cluster variables, namely the 9 canonical generators YMi , the 4 unipotent quantum minors YTi attached to the modules: T1 = 3 2 4 1 3 5 T2 = 3 2 4 1 3 5 2 T3 = 3 2 4 1 3 5 4 T4 = 3 2 4 1 3 5 2 4 and the elements YUi attached to the 2 modules U1 = 33 22 44 1 3 5 U2 = 3 2 4 1 33 5 2 4 with dimension vector α1 +2α2 +3α3 +2α4 +α5. These cluster variables form 50 clusters. 12.5 Quantum coordinate rings of open cells in partial flag varieties In this section, we assume that g is a simple Lie algebra of simply-laced type. We briefly review some classical material, using the notation of [GLS5, GLS6]. Let G be a simple simply connected complex algebraic group with Lie algebra g. Let H be a maximal torus of G, and B,B− a pair of opposite Borel subgroups containing H with unipotent radicals N,N−. We denote by xi(t) (i ∈ I, t ∈ C) the simple root subgroups of N, and by yi(t) the corresponding simple root subgroups of N−. We fix a non-empty subset J of I and we denote its complement by K = I \ J. Let BK be the standard parabolic subgroup of G generated by B and the one-parameter subgroups yk(t), (k ∈ K, t ∈ C). We denote by NK the unipotent radical of BK . Similarly, let B−K be the parabolic ... |

26 | T-systems and Y-systems in integrable systems
- Kuniba, Nakanishi, et al.
(Show Context)
Citation Context ... is a q-analogue (and an extension to the Kac-Moody case) of a determinantal identity of Fomin and Zelevinsky [FZ1]. We then transfer this identity to Aq(n) (Proposition 5.4). Here a little care must be taken since the restriction map from Aq(g) to Aq(n) is not a ring homomorphism. Appropriate specializations of this identity give rise, for every w ∈W , to a system Σw of equations (Proposition 5.5) allowing to calculate certain quantum minors depending on w in a recursive way. This system is a q-analogue of a T -system arising in various problems of mathematical physics and combinatorics (see [KNS]), and we believe it could be of independent interest. It will turn out that all quantum minors involved in Σw belong to the subalgebra Aq(n(w)), and that among them we find a set of algebra generators. On the other hand, we show that the generalized determinantal identities of [GLS6, Theorem 13.1], which relate a distinguished subset of cluster variables, take exactly the same form as the quantum T -system Σw when we lift them to the quantum cluster algebra AQ(q)(Cw). Therefore, after establishing that the quantum tori consisting of the initial variables of the two systems are isomorphic, we ... |

26 |
Multiplicative properties of dual canonical bases of quantum groups
- Reineke
- 1999
(Show Context)
Citation Context ...sis of Uq(n) adjoint to B with respect to the scalar product (·, ·). We call it the dual canonical basis of Uq(n), since it can be identified via Ψ with the dual basis of B in Aq(n). Note that B∗ is not invariant under the bar automorphism x 7→ x. The property of B∗ dual to (6.3) can be stated as follows. Let σ be the composition of the anti-automorphism ∗ and the bar involution, that is, σ is the ring anti-automorphism of Uq(n) such that σ(q) = q−1, σ(ei) = ei. (6.5) For β ∈ Q+, define N(β ) := (β , β ) 2 −degβ . (6.6) Then, if b ∈Uq(n)β belongs to B∗, there holds σ(b) = qN(β ) b, (6.7) (see [Re, Ki]). 6.3 Specialization at q = 1 of Uq(n) and Aq(n) Recall that UA(n) is the A-submodule of Uq(n) spanned by the canonical basis B. If we regard C as an A-module via the homomorphism q 7→ 1, we can define U1(n) := C⊗AUA(n). (6.8) This is a C-algebra isomorphic to the enveloping algebra U(n). Similarly, let AA(n) be the A-submodule of Aq(n) spanned by the basis Ψ(B∗). Define A1(n) := C⊗A AA(n). (6.9) This is a C-algebra isomorphic to the graded dual U(n)∗gr. This commutative ring can be identified with the coordinate ring C[N] of a pro-unipotent pro-group N with Lie algebra the completion n of n... |

24 | Auslander algebras and initial seeds for cluster algebras, - GEISS, LECLERC, et al. - 2007 |

22 | Kac-Moody groups and cluster algebras,
- GEISS, LECLERC, et al.
- 2011
(Show Context)
Citation Context ...a can be identified with the algebra C[N] of regular functions on the pro-unipotent group N attached to n. Dualizing Lusztig’s construction, one can obtain for each nilpotent representation X of Λ a distinguished regular function ϕX ∈ C[N], and the product ϕX ϕY can be calculated in terms of varieties of short exact sequences with end-terms X and Y [GLS2]. For each element w of the Weyl group W of g, the group N has a finite-dimensional subgroup N(w) of dimension equal to the length of w. In particular, when g is finite-dimensional, we have N = N(w0), where w0 is the longest element of W . In [GLS6], we have shown that the coordinate ring C[N(w)] is spanned by the functions ϕX where X goes over the objects of a certain subcategory Cw of mod(Λ). This category was introduced by Buan, Iyama, Reiten and Scott in [BIRS], and independently in [GLS4] for adaptable w. Moreover, we have proved that C[N(w)] has a cluster algebra structure in the sense of Fomin and Zelevinsky [FZ2], for which the cluster monomials are of the form ϕT for rigid objects T of Cw. The algebra C[N] has a quantum deformation, which we denote by Aq(n) (see below § 4.2), and it is well known that the algebras Uq(n) and Aq(n... |

22 | A quantum cluster algebra of Kronecker type and the dual canonical basis,
- LAMPE
- 2011
(Show Context)
Citation Context ...tion, obtained by q-deforming the Chevalley-Serre-type presentation of U(n). In contrast, quantum cluster algebras are defined as subalgebras of a skew field of rational functions in q-commuting variables, generated by a usually infinite number of elements given by an inductive procedure. As a matter of fact, there does not seem to be so many examples of “concrete” quantum cluster algebras in the literature. Grabowski and Launois [GL] have shown that the quantum coordinate rings of the Grassmannians Gr(2,n) (n ≥ 2), Gr(3,6), Gr(3,7), and Gr(3,8) have a quantum cluster algebra structure. Lampe [La1, La2] has proved two particular instances of Theorem 1.1, namely when g has type An or A (1) 1 and w = c 2 is the square of a Coxeter element. Recently, the existence of a quantum cluster structure on every algebra Aq(n(w)) was conjectured by Kimura [Ki, Conj.1.1]. Now Theorem 1.1 provides a large class of such examples, including all algebras Uq(n) for g of type A,D,E. By taking g = sln and some special permutation wk ∈ Sn, one also obtains that the quantum coordinate ring Aq(Mat(k,n− k)) of the space of k× (n− k)-matrices has a quantum cluster algebra structure for every 1 ≤ k ≤ n. This may be re... |

21 |
Braid group action and canonical bases,
- LUSZTIG
- 1996
(Show Context)
Citation Context ... k ≤ r). (7.14) 7.3 Action of Ti on unipotent quantum minors Proposition 7.1 Let λ ∈ P+, and u,v ∈ W be such that u(λ )− v(λ ) ∈ Q+, and consider the unipotent quantum minor du(λ ),v(λ ). Suppose that l(siu) = l(u)+1, and l(siv) = l(v)+1. Then Ti ( du(λ ),v(λ ) ) = (1−q−2)(αi,v(λ )−u(λ )) dsiu(λ ),siv(λ ). The proof will use Proposition 6.3 and the following lemmas. Lemma 7.2 We have Ti ◦ϕ = ϕ ◦Ti. Proof — This follows immediately from the definitions of ϕ and of Ti. 2 The next lemma is a restatement of a result of Kimura [Ki, Theorem 4.20], based on previous results of Saito [Sa] and Lusztig [Lu3]. Note that our Ti is denoted by T−1i in [Ki]. Lemma 7.3 Let b ∈B(∞) be such that εi(b) = 0. Then Ti(Gup(b)) = (1−q2)(αi,wt(b)) Gup ( f ϕ ∗ i (b) i (e ∗ i ) ε∗i (b)b ) , where ϕ∗i (b) := ε ∗ i (b)+(αi,wt(b)). Proof of Proposition 7.1 — By Lemma 7.2 and Proposition 6.3, we have ϕ ( Ti(du(λ ),v(λ )) ) = Ti ( Gup ( (e∗il(u)) bl(u) · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ )) . Let us write for short b := (e∗il(u)) bl(u) · · ·(e∗i1) b1 f cl(v) jl(v) · · · f c1j1 b∞ ∈B(∞). Then εi(b) = max{s |esi ·∆u(λ ),v(λ ) 6= 0}. The assumption l(siv) = l(v)+ 1 implies that (αi,v(λ )) ≥ 0, so by (3... |

18 | Quantum unipotent subgroup and dual canonical basis,
- KIMURA
- 2012
(Show Context)
Citation Context ...sis of Uq(n) adjoint to B with respect to the scalar product (·, ·). We call it the dual canonical basis of Uq(n), since it can be identified via Ψ with the dual basis of B in Aq(n). Note that B∗ is not invariant under the bar automorphism x 7→ x. The property of B∗ dual to (6.3) can be stated as follows. Let σ be the composition of the anti-automorphism ∗ and the bar involution, that is, σ is the ring anti-automorphism of Uq(n) such that σ(q) = q−1, σ(ei) = ei. (6.5) For β ∈ Q+, define N(β ) := (β , β ) 2 −degβ . (6.6) Then, if b ∈Uq(n)β belongs to B∗, there holds σ(b) = qN(β ) b, (6.7) (see [Re, Ki]). 6.3 Specialization at q = 1 of Uq(n) and Aq(n) Recall that UA(n) is the A-submodule of Uq(n) spanned by the canonical basis B. If we regard C as an A-module via the homomorphism q 7→ 1, we can define U1(n) := C⊗AUA(n). (6.8) This is a C-algebra isomorphic to the enveloping algebra U(n). Similarly, let AA(n) be the A-submodule of Aq(n) spanned by the basis Ψ(B∗). Define A1(n) := C⊗A AA(n). (6.9) This is a C-algebra isomorphic to the graded dual U(n)∗gr. This commutative ring can be identified with the coordinate ring C[N] of a pro-unipotent pro-group N with Lie algebra the completion n of n... |

15 |
Classical invariant theory for the quantum symplectic groups,
- STRICKLAND
- 1996
(Show Context)
Citation Context ... {4}. Then B−K\G is a smooth projective quadric Q of dimension 6, and NK can be regarded as an open cell O in Q. Here NK = N(w) where w = w0wK0 = s4s3s1s2s3s4. It is easy to check that the elements YMk (1≤ k ≤ 6) satisfy YM jYMi = qYMiYM j , (i < j, i+ j 6= 7), (12.18) YM4YM3 = YM3YM4 , (12.19) YM5YM2 = YM2YM5 +(q−q −1)YM3YM4 , (12.20) YM6YM1 = YM1YM6 +(q−q −1)(YM2YM5−q −1YM3YM4), (12.21) and that this is a presentation of the quantum coordinate ring Aq(O). This shows that Aq(O) is isomorphic to the quantum coordinate ring of the space of 4× 4-skew-symmetric matrices, introduced by Strickland [St]. The category Cw has finite representation type A1×A1. Hence, there are 4 cluster variables YM1 , YM2 , YM5 , YM6 , together with 4 frozen ones, namely, YV3 = YM3 , YV4 = YM4 , and YV5 = YM2YM5−q −1YM3YM4 , YV6 = YM1YM6−q −1YM2YM5 +q −2YM3YM4 . Observe that YV6 coincides with the quantum Pfaffian of [St]. There are 4 clusters {YM1 , YM2}, {YM1 , YM5}, {YM6 , YM2}, {YM6 , YM5}. Note that since all quantum cluster variables belong to the basis {YM(c) |c ∈Nr}, Conjecture 12.9 is easily verified in this case. Acknowledgments We thank the Hausdorff Center for Mathematics in Bonn for organizing a s... |

12 | Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases,
- GRABOWSKI, LAUNOIS
- 2011
(Show Context)
Citation Context ...tions of coordinate rings and quantizations of cluster algebras are defined in very different ways. For example Aq(n) ∼= Uq(n) is given by its Drinfeld-Jimbo presentation, obtained by q-deforming the Chevalley-Serre-type presentation of U(n). In contrast, quantum cluster algebras are defined as subalgebras of a skew field of rational functions in q-commuting variables, generated by a usually infinite number of elements given by an inductive procedure. As a matter of fact, there does not seem to be so many examples of “concrete” quantum cluster algebras in the literature. Grabowski and Launois [GL] have shown that the quantum coordinate rings of the Grassmannians Gr(2,n) (n ≥ 2), Gr(3,6), Gr(3,7), and Gr(3,8) have a quantum cluster algebra structure. Lampe [La1, La2] has proved two particular instances of Theorem 1.1, namely when g has type An or A (1) 1 and w = c 2 is the square of a Coxeter element. Recently, the existence of a quantum cluster structure on every algebra Aq(n(w)) was conjectured by Kimura [Ki, Conj.1.1]. Now Theorem 1.1 provides a large class of such examples, including all algebras Uq(n) for g of type A,D,E. By taking g = sln and some special permutation wk ∈ Sn, one ... |

12 |
Admissible Diagrams in Uwq (g) and Combinatoric Properties of Weyl Groups,
- MERIAUX, CAUCHON
- 2010
(Show Context)
Citation Context ...lusive, although Lampe [La1, La2] has proved it for all cluster variables in the two special cases mentioned above. Finally we note that it is well known that the algebras Aq(n(w)) are skew polynomial rings. Therefore, by Theorem 1.1, all quantum cluster algebras of the form AQ(q)(Cw) are also skew polynomial rings, which is far from obvious from their definition. One may hope that, conversely, the existence of a cluster structure on many familiar quantum coordinate rings will bring some new insights for studying their ring-theoretic properties, a very active subject in recent years (see e.g. [BG, GLL, MC, Y] and references therein). 3 2 The quantum coordinate ring Aq(g) 2.1 The quantum enveloping algebra Uq(g) Let g be a symmetric Kac-Moody algebra with Cartan subalgebra t. We follow the notation of [K2, §1]. In particular, we denote by I the indexing set of the simple roots αi (i ∈ I) of g, by P ⊂ t∗ its weight lattice, by hi(i ∈ I) the elements of P∗ ⊂ t such that 〈hi,α j〉= ai j are the entries of the generalized Cartan matrix of g. Since g is assumed to be symmetric, we also have a symmetric bilinear form (·, ·) on t∗ such that (αi,α j) = ai j. The Weyl group W < GL(t∗) is the Coxeter group ge... |

10 | Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves,
- GOODEARL, LAUNOIS, et al.
- 2011
(Show Context)
Citation Context ...lusive, although Lampe [La1, La2] has proved it for all cluster variables in the two special cases mentioned above. Finally we note that it is well known that the algebras Aq(n(w)) are skew polynomial rings. Therefore, by Theorem 1.1, all quantum cluster algebras of the form AQ(q)(Cw) are also skew polynomial rings, which is far from obvious from their definition. One may hope that, conversely, the existence of a cluster structure on many familiar quantum coordinate rings will bring some new insights for studying their ring-theoretic properties, a very active subject in recent years (see e.g. [BG, GLL, MC, Y] and references therein). 3 2 The quantum coordinate ring Aq(g) 2.1 The quantum enveloping algebra Uq(g) Let g be a symmetric Kac-Moody algebra with Cartan subalgebra t. We follow the notation of [K2, §1]. In particular, we denote by I the indexing set of the simple roots αi (i ∈ I) of g, by P ⊂ t∗ its weight lattice, by hi(i ∈ I) the elements of P∗ ⊂ t such that 〈hi,α j〉= ai j are the entries of the generalized Cartan matrix of g. Since g is assumed to be symmetric, we also have a symmetric bilinear form (·, ·) on t∗ such that (αi,α j) = ai j. The Weyl group W < GL(t∗) is the Coxeter group ge... |

8 | Quantum cluster algebras of type A and the dual canonical basis, arXiv:1101.0580,
- LAMPE
- 2010
(Show Context)
Citation Context ...tion, obtained by q-deforming the Chevalley-Serre-type presentation of U(n). In contrast, quantum cluster algebras are defined as subalgebras of a skew field of rational functions in q-commuting variables, generated by a usually infinite number of elements given by an inductive procedure. As a matter of fact, there does not seem to be so many examples of “concrete” quantum cluster algebras in the literature. Grabowski and Launois [GL] have shown that the quantum coordinate rings of the Grassmannians Gr(2,n) (n ≥ 2), Gr(3,6), Gr(3,7), and Gr(3,8) have a quantum cluster algebra structure. Lampe [La1, La2] has proved two particular instances of Theorem 1.1, namely when g has type An or A (1) 1 and w = c 2 is the square of a Coxeter element. Recently, the existence of a quantum cluster structure on every algebra Aq(n(w)) was conjectured by Kimura [Ki, Conj.1.1]. Now Theorem 1.1 provides a large class of such examples, including all algebras Uq(n) for g of type A,D,E. By taking g = sln and some special permutation wk ∈ Sn, one also obtains that the quantum coordinate ring Aq(Mat(k,n− k)) of the space of k× (n− k)-matrices has a quantum cluster algebra structure for every 1 ≤ k ≤ n. This may be re... |